Generative-enhanced optimization for knapsack problems: an industry-relevant study

TL;DR

This work addresses constrained industrial optimization by applying quantum-inspired tensor-network generative models to a generalized multi-knapsack problem. It develops two variants, TN-GEO and STN-GEO, leveraging MPS representations and $U(1)$ symmetry to encode equality constraints, alongside a DMRG-inspired training and perfect sampling to iteratively improve solution quality. Across 60 problem instances, the TN- and STN-GEO approaches achieve results comparable to simulated annealing, with hyper-parameter studies indicating that smaller bond dimensions and shorter training prevent overfitting and improve generalization. The study highlights practical considerations, such as symmetry maintenance, problem encoding choices, and ordering heuristics, while noting GEO’s higher computational cost relative to SA. Collectively, the results demonstrate the potential and limits of tensor-network–based generative optimization for industry-relevant combinatorial problems and point to directions for improving scalability and performance with quantum-classical hybrids.

Abstract

Optimization is a crucial task in various industries such as logistics, aviation, manufacturing, chemical, pharmaceutical, and insurance, where finding the best solution to a problem can result in significant cost savings and increased efficiency. Tensor networks (TNs) have gained prominence in recent years in modeling classical systems with quantum-inspired approaches. More recently, TN generative-enhanced optimization (TN-GEO) has been proposed as a strategy which uses generative modeling to efficiently sample valid solutions with respect to certain constraints of optimization problems. Moreover, it has been shown that symmetric TNs (STNs) can encode certain constraints of optimization problems, thus aiding in their solution process. In this work, we investigate the applicability of TN- and STN-GEO to an industry relevant problem class, a multi-knapsack problem, in which each object must be assigned to an available knapsack. We detail a prescription for practitioners to use the TN-and STN-GEO methodology and study its scaling behavior and dependence on its hyper-parameters. We benchmark 60 different problem instances and find that TN-GEO and STN-GEO produce results of similar quality to simulated annealing.

Figures (22)

• Figure 1: GEO pipeline. The process begins with the initial pairs of observations and their costs (step 0) and proceeds by selecting candidates for the training dataset (step 1) according to a selection strategy of choice. After selecting the candidates, the model computes a probability function $p(\mathbf{x})$ (step 2), followed by training the generative model (step 3). New samples are generated based on the trained model (step 4) and are then combined with the original training set (step 5). The updated training set is used for further iterations, continuing the loop for model refinement.
• Figure 2: Diagram of a Matrix Product State with $L=6$ elements.
• Figure 3: Schematic of the charge conservation for a symmetric tensor network. We use arrows to show incoming and outgoing indices $k_i$, with possible values of the charge written in gray on top of each arrow. Dashed arrows display incoming particle number $b$ and outgoing $0$, which follows from Eq. \ref{['eq:charge_conservation']}.
• Figure 4: Schematic of the charge conservation in a MPS with binary encoding for a knapsack problem with $N=2$ objects and $M=3$ knapsacks. Each $x_{ij}$ is mapped to an index of the physical leg of $T^{[M(i-1)+j]}$, and the cardinality constraint reads $\sum_{j=1}^{M} x_{ij} = 1$. The incoming charge vector changes based on the physical leg index at each tensor, preserving the particle. The numbers $0$ and $1$ above the vertical leg correspond to physical index values, showing their influence on the charge vector by adjusting the charge value to their right accordingly. The indices and charge vectors in red demonstrate and example of assigning the $1$st object to the $2$nd knapsack, and the $2$nd object to the $3$rd knapsack. Black font denotes other possible indices and charge vectors unrelated to the example. Dashed arrows denote dimensions equal to $1$.
• Figure 5: MPS with integer encoding for a knapsack problem with $N=2$ objects and $M=3$ knapsacks. Red color denotes an example of assigining the $1$st object to the $2$nd knapsack, and the $2$nd object to the $3$rd knapsack, i.e. $y=(2, 3)^\top$.